The geometric mean of two positive integers and is . In this post, we are going to show the relationship between geometric mean and the relationships among the sides and altitude of a right triangle.

The Right Triangle Altitude Theorem

In a right triangle,

the altitude to the hypotenuse is the geometric mean of the segments into which it divides (the hypotenuse); and

each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

The two theorems above, states that in the triangle below,

, and .

Proof

Let and be the legs of the triangle with hypotenuse . Let be the altitude of the triangle to the hypotenuse dividing the to segments and .

First, we show that that the three triangles, , , and are similar triangles.

since both of them equal . So, , , and are right triangles which all include an angle whose measure equal . Therefore, these three triangles are similar. Since these three triangles are similar, their sides are proportional.